Abstract

This thesis describes an investigation into the conceptualunderstanding of algebra by early learners (age 11-13 years) andhow a computer-based approach may be used to improve such,without any consequent loss of manipulative skills.The psychological framework for the investigation centred on theimportance of the individual child's construction of a cognitiveframework of knowledge and the relevance of the current state ofthis to the facilitating of concept acquisition. As such itincorporates elements of the developmental psychology of Piaget,Ausubel and Skemp. Furthermore, in order to assist in thesynthesis of a sufficiently broad psychological theory ofeducation it was necessary to postulate the formulation of a newintegrated bi-modal model of learning. This is described, alongwith details of its application and significance to a theory ofcognitive integration which is designed to promote versatilelearning (after Brumby, 1982) in mathematics through arelational linking of global/holistic and serialist/analyticschemas.

The research comprised two initial investigations followed bythe main experiment. The results of the initial investigationswith early learners of algebra showed that the dynamic algebramodule written for the research produced a significantimprovement in the children's conceptual understanding ofalgebra. The main experiment sought to further clarify thisimprovement and to compare and contrast it with that produced bya traditional skill-based algebra module. In order to facilitatethis comparison, the performance of 57 matched pairs of pupilsfrom two groups of three parallel forms of the first year of a12+ entry co-educational secondary school was analysed.

The results of the investigation confirmed the value of thedynamic algebra module as a generic organiser (in thesense of Tall, 1986) for the understanding of algebraicconcepts, producing a significant difference in conceptualunderstanding, without any detrimental effect on manipulativeskills. Furthermore, the beneficial effects of the programmewere such that its results showed that it had provided a betterbase than the skill-based approach for the extension ofalgebraic understanding past the initial stages and into moreinvolved areas such as linear equations and inequalities.

The findings of this research show that the use of amodule based on a computer environment, with its manyadvantages for conceptual learning, prior to the more formalintroduction of algebraic techniques, is of great cognitivevalue. They also provide evidence for the theoretical model oflearning proposed in the thesis, and suggest that for theproduction of a versatile learner in mathematics, more attentionshould be paid to the integration of the global/holisticabilities of the individual with his/her serialist/analyticabilities. The implications for the future are that suchabilities, and hence mathematical competence may well beimproved in other areas of the curriculum by the use of thecomputer within a similar theoretical framework.